Reminder: killing a group is gote!

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xela
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Reminder: killing a group is gote!

Post by xela »

This interesting variation came up, reviewing one of my games with KataGo:
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | . X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
How should black respond to :w1:?

Hint:
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | 2 X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
Not here! And not anywhere else at the bottom left.
What KataGo showed me:
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . 2 . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O a . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | . X X X O O O . . 0 . . . . . X . . . |
$$ | 6 5 . 4 X X 1 . b . . . . . . . . . . |
$$ | . 7 9 . 8 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
If :b2: (or black at a, which is about equally good), and if white goes on to kill the corner, then black comes away with sente, and has a useful forcing move at b later. Killing the corner was only worth about a 20 point swing (or miai value 10 points): the combination of :b2: and :b10: is more than enough to make up for it.

The move :w1: itself is KataGo's suggestion: even though it's gote, it's very big, both for the outside influence, and for the option of taking the corner later at the right time.
(And to save anyone else saying it: yes, black should have played at :w1: a fair bit earlier.)
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Gérard TAILLE
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Re: Reminder: killing a group is gote!

Post by Gérard TAILLE »

xela wrote:This interesting variation came up, reviewing one of my games with KataGo:
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | . X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
How should black respond to :w1:?

Hint:
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | 2 X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
Not here! And not anywhere else at the bottom left.
What KataGo showed me:
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . 2 . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O a . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | . X X X O O O . . 0 . . . . . X . . . |
$$ | 6 5 . 4 X X 1 . b . . . . . . . . . . |
$$ | . 7 9 . 8 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
If :b2: (or black at a, which is about equally good), and if white goes on to kill the corner, then black comes away with sente, and has a useful forcing move at b later. Killing the corner was only worth about a 20 point swing (or miai value 10 points): the combination of :b2: and :b10: is more than enough to make up for it.

The move :w1: itself is KataGo's suggestion: even though it's gote, it's very big, both for the outside influence, and for the option of taking the corner later at the right time.
(And to save anyone else saying it: yes, black should have played at :w1: a fair bit earlier.)
Your question is very interesting and I have difficulty to find good arguments
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | 2 X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . 4 . 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
Assume :b2: is a bad move because the value of the corner is not that big.
In this case it seems to me that the two folowing moves :w3: and :b4: should also be bad for the same reason. Can you conirm that these moves :w3: and :b4: are not in the top moves of katago?
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dust
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Re: Reminder: killing a group is gote!

Post by dust »

Just wondering how you reached this position (which looks slightly odd in the lower left corner). Did the game proceed like this?

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pwaldron
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Re: Reminder: killing a group is gote!

Post by pwaldron »

Gérard TAILLE wrote: Your question is very interesting and I have difficulty to find good arguments
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | a O O O . . . . . , . . . . . . . . . |
$$ | 2 X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . 4 . 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
Assume :b2: is a bad move because the value of the corner is not that big.
In this case it seems to me that the two folowing moves :w3: and :b4: should also be bad for the same reason. Can you conirm that these moves :w3: and :b4: are not in the top moves of katago?
I had the same thought, but I note 'a' also creates a living shape, and has more impact on white's left side.
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dfan
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Re: Reminder: killing a group is gote!

Post by dfan »

Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | 2 X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . 4 . 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
Assume :b2: is a bad move because the value of the corner is not that big.
In this case it seems to me that the two folowing moves :w3: and :b4: should also be bad for the same reason. Can you conirm that these moves :w3: and :b4: are not in the top moves of katago?
When I looked with KataGo, :w3: was the second-best move, slightly behind the 3-3 invasion at the top right. If White does play :w3:, it thinks that Black must play :b4:, and if Black tenukis instead, White should cut and kill.

I think it is important in assessing the position after :b2: that Black has invested an extra stone (and an extra move!!) in the bottom left. If White kills the group, then compared to before :b2:, they get an extra point, and Black has wasted a whole turn; if Black lives, then compared to before :b2:, they get some extra endgame points or an invasion on the left side gets some more opportunities.
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Gérard TAILLE
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Re: Reminder: killing a group is gote!

Post by Gérard TAILLE »

dfan wrote:
Gérard TAILLE wrote:
I think it is important in assessing the position after :b2: that Black has invested an extra stone (and an extra move!!) in the bottom left. If White kills the group, then compared to before :b2:, they get an extra point, and Black has wasted a whole turn; if Black lives, then compared to before :b2:, they get some extra endgame points or an invasion on the left side gets some more opportunities.
I do not understand where black has invested an extra stone.
What is the difference between white killing black after :w1: or killing black after :w1: :b2: :w3: ?
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dfan
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Re: Reminder: killing a group is gote!

Post by dfan »

I may be misunderstanding your original question.

You said: "Assume :b2: is a bad move because the value of the corner is not that big", and then seemed to deduce that any subsequent moves in that corner must be similarly bad.

I'm saying that the value of the corner is bigger after :b2: than before it. (In the following two diagrams, :w3: is bigger in the second diagram than the first.)

So I don't think there is a contradiction between
  • After :w1:, the corner is not big enough (right now) to fight over (so :b2: in the second diagram below is a bad move).
  • After :b2: the corner is big enough to fight over (so :w3: in the second diagram is a good move).
But you are a stronger player than I am so I may be missing something.
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . 2 . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | . X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | 2 X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
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xela
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Re: Reminder: killing a group is gote!

Post by xela »

dust wrote:Just wondering how you reached this position (which looks slightly odd in the lower left corner). Did the game proceed like this?
Not quite.

OK, I may as well share the full game in all its ugliness. Joseki mistakes were made in other corners too.
I'm still not sure if I enjoy these fast, messy online games. But opportunities to play over a board are a bit limited here. Looking forward to my US trip...

Attachments
killing_is_gote.sgf
(1.58 KiB) Downloaded 590 times
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xela
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Re: Reminder: killing a group is gote!

Post by xela »

dfan wrote:
Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$Wc19
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | 2 X X X O O O . . . . . . . . X . . . |
$$ | . . . a X X 1 . c . . . . . . . . . . |
$$ | . . . 4 . 3 b . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
Assume :b2: is a bad move because the value of the corner is not that big.
In this case it seems to me that the two folowing moves :w3: and :b4: should also be bad for the same reason. Can you confirm that these moves :w3: and :b4: are not in the top moves of katago?[/hide]
When I looked with KataGo, :w3: was the second-best move, slightly behind the 3-3 invasion at the top right. If White does play :w3:, it thinks that Black must play :b4:, and if Black tenukis instead, White should cut and kill.

I think it is important in assessing the position after :b2: that Black has invested an extra stone (and an extra move!!) in the bottom left. If White kills the group, then compared to before :b2:, they get an extra point, and Black has wasted a whole turn; if Black lives, then compared to before :b2:, they get some extra endgame points or an invasion on the left side gets some more opportunities.[/hide]
If black leaves out :b4: and white kills with a, there's an important difference from the killing sequence shown earlier on this page. But it's nothing to do with the local tally (the exchange :b2: - :w3: is one stone for each side). The difference is that black no longer has the threat of escaping at b, and so the point c is no longer a forcing move for black. So killing this way is more profitable for white, compared to the previous sequence.
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Gérard TAILLE
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Re: Reminder: killing a group is gote!

Post by Gérard TAILLE »

Click Here To Show Diagram Code
[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . 2 . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | . X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]
May be I am wrong but if I understand correctly you claim that, according to katago, the above sequence is not bad for both.
Typically that means that the miai value of each o these three moves would be about 13 points.
Now my difficulty is the following: if :b2: is consider a bad move in the bottom left corner why this white :w3: can be considered a good move?
IOW why the arguments saying that :b2: is a bad move in the bottom left corner are not valid when analysing the value of the :w3: move in the same corner? If the corner is small for black, could it be big enough for white?
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xela
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Re: Reminder: killing a group is gote!

Post by xela »

Sorry, I wasn't clear. In my first post, I said "If :b2: ... and if white goes on to kill the corner..." This was my way of hinting that KataGo does not think :w3: is the best move. It prefers to play in the bottom right. I just put :w3: on the board to see what KataGo thought of it, and then it "explained" to me that capturing the corner is not so good for white, because it leaves some aji for black.
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Gérard TAILLE
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Re: Reminder: killing a group is gote!

Post by Gérard TAILLE »

I am trying to convinced me that defending the corner with black :b2: is not good at this stage of the game, but it is still difficult.

What is my reasonning?

Firstly I imagine that the status of the black has no impact on the outside as in the following diagram
Click Here To Show Diagram Code
[go]$$B
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . . . |
$$ | . O O O a . . . . , . . . . . . . . . |
$$ | . X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X O O O . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ +---------------------------------------+[/go]
If black lives in gote I imagine that the very final result could be something like (I do not look for a very accurate value):
Click Here To Show Diagram Code
[go]$$B
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . . . |
$$ | 6 O O O . . . . . , . . . . . . . . . |
$$ | 5 X X X O O O . . . . . . . . X . . . |
$$ | . . 1 . X X O O O . . . . . . . . . . |
$$ | . . 7 . 3 2 4 . O . . . . . . . . . . |
$$ +---------------------------------------+[/go]
and I see a swing value about equal to 28 points (miai value 14).

Secondly comes the crucial point. How to evaluate the impact of black group on the outside?
For me the position is quite difficult because three situations exist:
1) Black plays first in the corner in order to live
2) White plays first in the corner in order to kill black
3) Neither player play in the corner
In each case the influence of the corner on the outside is not easy to evaluate but general comment may be formulated:
In situation 1) I imagine the influence on the outside is in white favor because white may have outside moves renforcing white and threatening black corner
In situation 2) I imagine the influence on the outside is in black favor because black may have outside moves threatening to connect the killed black group
In siutation 3) I imagine the influence on the outside is still more in black favor due to the potential cut at "a": if black can cut at "a" threatening to attack one of the white groups, then white can hardly sacrify one of her groups because black stones in the corner will live without having added a defensive stone.

The conclusion is not easy is it?
At least I note a very general point. When you count the gain for living or for killing a group, and ignoring the influence on the outside, then you get always an overevaluation.

Here, prefering to play tenuki (about a move gaining 13 points) instead of gaining 14 points in the corner means that the influence on the outside must be more than 2 points : (14-13)*2
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